Answer: 1 - C
2 - E
3 - no answer
4 - B
Step-by-step explanation:
A. ![y = 3x+x^2](https://tex.z-dn.net/?f=y%20%3D%203x%2Bx%5E2)
![y' = 3 + 2x\\y'' = 2](https://tex.z-dn.net/?f=y%27%20%3D%203%20%2B%202x%5C%5Cy%27%27%20%3D%202)
![y'' + y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%20y%20%3D%200)
![2 + 3x + x^2 \neq 0](https://tex.z-dn.net/?f=2%20%2B%203x%20%2B%20x%5E2%20%5Cneq%200)
So, A is not an answer for 1
![y' = 3y](https://tex.z-dn.net/?f=y%27%20%3D%203y)
![3+2x = 3(3x + x^2)](https://tex.z-dn.net/?f=3%2B2x%20%3D%203%283x%20%2B%20x%5E2%29)
So, A is not an answer for 2
![2x^2y'' + 3xy'= y](https://tex.z-dn.net/?f=2x%5E2y%27%27%20%2B%203xy%27%3D%20y)
![2x^2(2) + 3x(3 + 2x) = 4x^2 + 9x + 6x^2 \neq 3x+x^2](https://tex.z-dn.net/?f=2x%5E2%282%29%20%2B%203x%283%20%2B%202x%29%20%3D%204x%5E2%20%2B%209x%20%2B%206x%5E2%20%5Cneq%203x%2Bx%5E2)
So, A is not an answer for 3
![y'' + 6y' + 8y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%206y%27%20%2B%208y%20%3D%200)
![2 + 6(3+2x)+8(3x + x^2) = 2+18+12x+24x+8x^2 \neq 0](https://tex.z-dn.net/?f=2%20%2B%206%283%2B2x%29%2B8%283x%20%2B%20x%5E2%29%20%3D%202%2B18%2B12x%2B24x%2B8x%5E2%20%5Cneq%200)
So, A is not an answer for 4
B. ![y = e^{-4x}](https://tex.z-dn.net/?f=y%20%3D%20e%5E%7B-4x%7D)
![y' = -4e^{-4x}](https://tex.z-dn.net/?f=y%27%20%3D%20-4e%5E%7B-4x%7D)
![y'' = 16e^{-4x}](https://tex.z-dn.net/?f=y%27%27%20%3D%2016e%5E%7B-4x%7D)
![y'' + y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%20y%20%3D%200)
![16e^{-4x} -4e^{-4x} = 12e^{-4x} \neq 0](https://tex.z-dn.net/?f=16e%5E%7B-4x%7D%20-4e%5E%7B-4x%7D%20%3D%2012e%5E%7B-4x%7D%20%5Cneq%200)
So, B is not an answer for 1
![y' = 3y](https://tex.z-dn.net/?f=y%27%20%3D%203y)
![-4e^{-4x} \neq 3e^{-4x}](https://tex.z-dn.net/?f=-4e%5E%7B-4x%7D%20%5Cneq%203e%5E%7B-4x%7D)
So, B is not an answer for 2
![2x^2y'' + 3xy' = y](https://tex.z-dn.net/?f=2x%5E2y%27%27%20%2B%203xy%27%20%3D%20y)
![2x^2(16e^{-4x}) +3x(-4e^{-4x}) \neq e^{-4x}](https://tex.z-dn.net/?f=2x%5E2%2816e%5E%7B-4x%7D%29%20%2B3x%28-4e%5E%7B-4x%7D%29%20%5Cneq%20%20e%5E%7B-4x%7D)
So, B is not an answer for 3
![y'' + 6y' +8y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%206y%27%20%2B8y%20%3D%200)
![16e^{-4x} + 6(-4e^{-4x}) + 8e^{-4x} = e^{-4x}(16-24+8) = 0](https://tex.z-dn.net/?f=16e%5E%7B-4x%7D%20%2B%206%28-4e%5E%7B-4x%7D%29%20%2B%208e%5E%7B-4x%7D%20%3D%20e%5E%7B-4x%7D%2816-24%2B8%29%20%3D%200)
So, B is an answer for 4
C. ![y = sin(x)](https://tex.z-dn.net/?f=y%20%3D%20sin%28x%29)
![y' = cos(x)](https://tex.z-dn.net/?f=y%27%20%3D%20cos%28x%29)
![y'' = -sin(x)](https://tex.z-dn.net/?f=y%27%27%20%3D%20-sin%28x%29)
![y'' + y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%20y%20%3D%200)
![-sin(x) + sin(x) = 0](https://tex.z-dn.net/?f=-sin%28x%29%20%2B%20sin%28x%29%20%3D%200)
So, C is an answer for 1
We jump to
D. ![y = x^{12}](https://tex.z-dn.net/?f=y%20%3D%20x%5E%7B12%7D)
![y' = 12x^{11}](https://tex.z-dn.net/?f=y%27%20%3D%2012x%5E%7B11%7D)
![y'' = 132 x^{10}](https://tex.z-dn.net/?f=y%27%27%20%3D%20132%20x%5E%7B10%7D)
![y' = 3y](https://tex.z-dn.net/?f=y%27%20%3D%203y)
![12x^{11} \neq 3x^{12}](https://tex.z-dn.net/?f=12x%5E%7B11%7D%20%5Cneq%203x%5E%7B12%7D)
So, D is not an answer for 2
![2x^2y'' + 3xy' = y](https://tex.z-dn.net/?f=2x%5E2y%27%27%20%2B%203xy%27%20%3D%20y)
![2x^2(132x^{10}) + 3x(12x^{11}) = 300x^{12} \neq x^{12}](https://tex.z-dn.net/?f=2x%5E2%28132x%5E%7B10%7D%29%20%2B%203x%2812x%5E%7B11%7D%29%20%3D%20300x%5E%7B12%7D%20%5Cneq%20x%5E%7B12%7D)
So, D is not an answer for 3
E. ![y = 6e^{3x}](https://tex.z-dn.net/?f=y%20%3D%206e%5E%7B3x%7D)
![y' = 18e^{3x}](https://tex.z-dn.net/?f=y%27%20%3D%2018e%5E%7B3x%7D)
![y'' = 54e^{3x}](https://tex.z-dn.net/?f=y%27%27%20%3D%2054e%5E%7B3x%7D)
![y' = 3y](https://tex.z-dn.net/?f=y%27%20%3D%203y)
![18e^{3x} = 3(6e^{3x}) = 18e^{3x}](https://tex.z-dn.net/?f=18e%5E%7B3x%7D%20%3D%203%286e%5E%7B3x%7D%29%20%3D%2018e%5E%7B3x%7D)
So, E is an answer for 2